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COMMON TERMS IN MATHEMATICS
Dilara DORAK & M.Tevfik
DORAK
Please bookmark
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See also
'Common
Concepts in Statistics', 'Glossary
of Mathematical Mistakes' and 'CTK Glossary of Mathematical
Terms’
[Please note that the best way to
find an entry is to use the Find option from the Edit menu, or CTRL + F]
Absolute
value: The magnitude of a number. It is the
number with the sign (+ or -) removed and is symbolised using two vertical straight
lines ( |5| ). Also called modulus.
Abstract
number: A number with no associated units.
Acute angle:
An angle with degree measure less than 90. See MathWorld:
Geometry:
Trigonometry:
Angles.
Addition:
The process of finding the sum of two numbers, which are called addend
and the augend (sometimes both
are called the addend).
Algorithm:
Any mathematical procedure or instructions involving a set of steps to solve a
problem.
Arctan:
The inverse of the trigonometric function tangent shown as arctan(x) or tan-1(x). It is useful in vector
conversions and calculations. See Wikipedia: Mathematics: Trigonometric
Functions.
Arithmetic mean:
M = (x1 + x2 + .... xn) / n
(n = sample size).
Arithmetic sequence:
A sequence of numbers in which each term (subsequent to the first) is generated
by adding a fixed constant to its predecessor.
Associative
property: A binary operation (*) is defined associative if, for
a*(b*c) = (a*b)*c. For example, the operations addition and multiplication of
natural numbers are associative, but subtraction and division are not.
Asymptote:
A straight line that a curve approaches but never meets or crosses. The curve
is said to meet the asymptote at infinity. In the equation y = 1/x, y becomes
infinitely small as x increases but never reaches zero.
Axiom:
Any assumption on which a mathematical theory is based.
Average:
The sum of several quantities divided by the number of quantities.
Avogadro's number:
The number of molecules in one mole is called Avogadro’s number
(approximately 6.022 × 1023 particles/mole).
Binary operation:
An operation that is performed on just two elements of a set at a time.
Brownian motion: See an article (by Lee & Hoon) and an animation.
Butterfly effect:
In a system when a small change results in an unpredictable and
disproportionate disturbance, the effect causing this is called a butterfly
effect.
Calculus:
Branch of mathematics concerned with rates of change, gradients of curves,
maximum and minimum values of functions, and the calculation of lengths, areas and
volumes. It involves determining areas (integration) and tangents
(differentiation), which are mutually inverse. Also called real
analysis. See also Dr. Vogel's
Gallery of Calculus Pathologies; MathWorld:
Calculus;
Wikipedia: Mathematics: Calculus;
Visual
Calculus; Math Archives: Calculus; Calculus
Animations with Mathcad.
Cartesian
coordinates: Cartesian coordinates (x,y) specify
the position of a point in a plane relative to the horizontal x and the
vertical y axes. The x and y axes form the basis of two-dimensional Cartesian
coordinate system.
Chaos: Apparent randomness whose origins are entirely
deterministic. A state of disorder and irregularity whose evolution in time,
though governed by simple exact laws, is highly sensitive to starting
conditions: a small variation in these conditions will produce wildly different
results, so that long-term behaviour of chaotic systems cannot be predicted.
This sensitivity to initial conditions is also known as the butterfly
effect (when a butterfly flaps its wings in
Mexico, the result may be a hurricane in Florida a month later).
Chord: A straight line joining two points on a curve or a
circle. See also secant line.
Circle: A circle is defined as the set of points at a
given distance (or radius) from its centre. If the coordinates of the centre of
a circle on a plane is (a,b) and the radius is r, then (x-a)2 +
(y-b)2 = r2. The equation that characterises a circle has
the same coefficients for x2 and y2. The area of a circle
is A = pr2 and circumference is C = 2pr. A circle with centre (a,b) and radius r has parametric equations: x
= a + r.cos q and y = b +
r.sin q (0 ≤
q ≤ 2p). A
‘tangent’ is a line, which touches a circle at one point (called
the point of tangency) only. A ‘normal’ is a line, which goes
through the centre of a circle and through the point of tangency (the normal is
always perpendicular to the tangent). A straight line can be considered a
circle; a circle with infinite radius and centre at infinity. See a Lecture
Note, BBC
Bitesize: Circle; Wikipedia: Mathematics: Circle; MathWorld: Geometry: Circles.
Circumference: A line or boundary that forms the perimeter of a
circle.
Closure
property: If the result of doing an operation
on any two elements of a set is always an element of the set, then the set is
closed under the operation. For example, the operations addition and
multiplication of natural numbers (the set) are closed, but subtraction and
division are not.
Coefficient: A number or letter before a variable in an
algebraic expression that is used as a multiplier.
Common
denominator: A denominator that is common to
all the fractions within an equation. The smallest number that is a common
multiple of the denominators of two or more fractions is the lowest (or
least) common denominator (LCM).
Common
factor: A whole number that divides exactly
into two or more given numbers. The largest common factor for two or more
numbers is their highest common factor
(HCF).
Common
logarithm: Logarithm with a base of 10 shown
as log10 [log1010x = x].
Common
ratio: In a geometric sequence, any term
divided by the previous one gives the same common ratio.
Commutative
property: A binary operation (*) defined on a
set has the commutative property if for every two elements, a and b, a*b = b*a.
For example, the operations addition and multiplication of natural numbers are
commutative, but subtraction and division are not.
Complementary
angles: Two angles whose sum is 90o. See
also supplementary angles.
Complex
numbers: A combination of real and imaginary
numbers of the form a + bi where a and b are real numbers
and i is the square root of -1 (see imaginary
number). While real numbers can be represented
as points on a line, complex numbers can only be located on a plane. See Types
of Numbers.
Composite
number: Any integer which is not a prime number,
i.e., evenly divisible by numbers other than 1 and itself.
Congruent: Alike in all relevant respects.
Constant: A quality of a measurement that never changes in
magnitude.
Coordinate: A set of numbers that locates the position of a
point usually represented by (x,y) values.
Cosine
law: For any triangle, the side lengths a, b,
c and corresponding opposite angles A, B, C are related as follows: a2
= b2 + c2 - 2bc cosA etc. The law of cosines is useful to
determine the unknown data of a triangle if two sides and an angle are known. See
Wikipedia: Cosine
Law.
Counting
number: An element of the set C = {1,2,3,...}.
Cube
root: The factor of a number that, when it is
cubed (i.e., x3) gives that number.
Curve: A line that is continuously bent.
Decimal: A fraction having a power of ten as denominator,
such as 0.34 = 34/100 (102) or 0.344 = 344/1000 (103). In
the continent, a comma is used as the decimal point (between the unit figure
and the numerator).
Degree
of an angle: A unit of angle equal to one
ninetieth of a right angle. Each degree ( 0 ) may be further
subdivided into 60 parts, called minutes (60’),
and in turn each minute may be subdivided into another 60 parts, called seconds (60’’). Different
types of angles are called acute (<900)< right (900)
< obtuse (900-1800) < reflex (1800-3600).
See also radian (the SI unit of angle).
Denominator: The bottom number in a fraction.
Derivative: The derivative at a point on a curve is the
gradient of the tangent to the curve at the given point. More technically, a
function (f'(x0)) of a function y = f(x),
representing the rate of change of y and the
gradient of the graph at the point where x = x0, usually shown as dy/dx. The notation dy/dx suggests the ratio of two numbers dy and dx (denoting
infinitesimal changes in y and x), but it is a single number, the limit of a ratio
(k/h) as they both approach zero. Differentiation is the process of calculating
derivatives. The derivatives of all commonly occurring functions are known. See
Function,
Derivative & Integral Applet; Calculus Graphics; Mathlets: Derivative
Calculator.
Differential
Equations: Equations containing one or more
derivatives (rate of change). As such these equations represent the
relationships between the rates of change of continuously varying quantities.
The solution contains constant terms (constant of integration) that are not
present in the original differential equation. Two general types of
differential equations are ordinary differential equations (ODE) and partial
differential equations (PDE). When the function involved in the equation
depends upon only a single variable, the differential equation is an ODE. If
the function depends on several independent variables (so that its derivatives
are partial derivatives) then the differential equation is a PDE. See Internet
Resources for Differential Equations.
Extrapolation: Estimating the value of a function or a quantity
outside a known range of values.
Interpolation: Estimating the value of a function or a quantity
from known values on either side of it.
Inverse
function: A function which 'does
the reverse' of a given function. For example, functions with the prefix arc
are inverse trigonometric functions; e.g. arcsin x for the inverse of
sin(x).See also Wikipedia: Mathematics:
Inverse
Functions and Logarithmic Inverse Functions.
Diameter: A straight line that passes from side to side
thorough the centre of a circle.
Differential
calculus: Differentiation is concerned with
rates of change and calculating the gradient at any point from the equation of
the curve, y =
f(x).
Differential
equation: Equations involving total or partial
differentiation coefficients and the rate of change; the difference between
some quantity now and its value an instant into the future. See also Wikipedia: Mathematics: Differential Equations; Differential Equations Applets.
Digit: In the decimal system, the numbers 0 through 9.
Dimension: Either the length and/or width of a flat surface
(two-dimensional); or the length, width, and/or height of a solid
(three-dimensional).
Distributive
property: A binary operation (*) is
distributive over another binary operation (^) if, a*(b^c) = (a*b)^(a*c). For
example, the operation of multiplication is distributive over the operations of
addition and subtraction in the set of natural numbers.
Division: The operation of ascertaining how many times one
number, the divisor, is contained in another, the dividend. The result is the quotient, and any number left over is called the remainder. The dividend and divisor are also called the numerator and denominator,
respectively.
Dynamics: The branch of mathematics, which studies the way
in which force produces motion.
e: Symbol for the base of natural
logarithms (2.7182818285...), defined as the limiting value of (1 + 1/m)m.
Equilibrium: The state of balance between opposing forces or
effects.
Even
number: A natural number that is divisible by
two.
Exponent (power, index): A number denoted by a small numeral placed above
and to the right of a numerical quantity, which indicates the number of times
that quantity is multiplied by itself. In the case of Xn, it
is said that X is raised to the power of n. When a and b are non-zero real
numbers and p and q are integers, the following rules of power apply:
ap
x aq
= ap+q; (ap)q
= apq; (a1/n)m
= am/n; a1/2 x b1/2 =
(ab)1/2.
Exponential
function: A function in the form of f(x) = ax
where x is a real number, and a is positive and not 1. One exponential function
is f(x) = ex.
Factorial: The product of a series of consecutive positive
integers from 1 to a given number (n). It is expressed with the symbol ( ! ).
For example, 5! = 5x4x3x2x1 = 120. As a rule (n!+n) is evenly divisible by n.
Factor: When two or more natural numbers are multiplied,
each of the numbers is a factor of the product. A factor is then a number by
which another number is exactly divided (a divisor) .
Factorisation: Writing a number as the product of its factors
which are prime numbers.
Fermat's
little theorem: If p is a prime number and b
is any whole number, then bp-b is a multiple of p (23 - 2
= 6 and is divisible by 3).
Fermat
prime: Any prime number in the form of 22n + 1
(see also Mersenne prime).
Fibonacci
sequence: Sequence of integers, where each is
the sum of the two preceding it. 1,1,2,3,5,8,13,21,... The number of petals of
flowers forms a Fibonacci series.
Fractals: Geometrical entities characterised by basic
patterns that are repeated at ever decreasing sizes. They are relevant to any
system involving self-similarity repeated on diminished scales (such as a
fern's structure) as in the study of chaos.
Fraction
(quotient): A portion of a whole amount. The
term usually applies only to ratios of integers (like 2/3, 5/7). Fractions less
than one are called common, proper or vulgar fractions; and those greater than 1 are called improper fraction.
Function
(f): The mathematical operation that
transforms a piece of data into a different one. For example, f(x) = x2 is a function transforming any number to its
square.
Geometry in Wikipedia.
Geometric
mean: G = (x1.x2...xn)1/n
where n is the sample size. This can also be expressed as antilog ((1/n) S log x). See Applications
of the Geometric Mean.
Geometric
sequence: A sequence of numbers in which each
term subsequent to the first is generated by multiplying its predecessor by a fixed
constant (the common ratio).
Goldbach
conjecture: Every even number greater than 4
is the sum of two odd primes (32 = 13 + 19). Every odd number greater than 7
can be expressed as the sum of three odd prime numbers (11 = 3 + 3 + 5).
Gradient: The slope of a line. The gradient of two points on
a line is calculated as rise (vertical increase) divided by run (horizontal
increase), therefore, the gradient of a line is equal to the tangent of the
angle it makes with the positive x-axis (y/x). See Curve
Bank: Slope.
Greek
alphabet: For list of Greek letters follow the
link.
Harmonic
mean: Of a set of numbers (y1 to yi),
the harmonic mean is the reciprocal of the arithmetic mean of the reciprocal of
the numbers [H = N / S (1/y)]. See also Wikipedia: Mathematics: Harmonic
Mean.
Hierarchy
of operations: In an equation with multiple
operators, operations proceed in the following order: (brackets),
exponentiation, division/multiplication, subtraction/summation and from left to
right.
Highest
common factor (HCF): The greatest natural
number, which is a factor of two or more given numbers.
Hypotenuse: The longest side of a right triangle, which lies
opposite the vertex of the right angle.
i: The square root of -1 (an imaginary number).
Identity
element: The element of a set which when
combined with any element of the same set leaves the other element unchanged
(like zero in addition and subtraction, and 1 in multiplication or division).
Imaginary
number: The product of a real number x and i, where i2 + 1 =
0. A complex number in which the real part is zero. In general, imaginary
numbers are the square roots of negative numbers. See Types
of Numbers.
Improper
fraction: A fraction whose numerator is the
same as or larger than the denominator; i.e., a fraction equal to or greater
than 1.
Infinite: Having no end or limits. Larger than any
quantified concept. For many purposes it may be considered as the reciprocal of
zero and shown as an 8 lying on its side (∞).
Infinitesimal: A vanishingly small part of a quantity. It equals
almost zero.
Integer: Any whole number: positive and negative whole numbers
and zero.
Integral
calculus: This is the inverse process to
differentiation; i.e., a function which has a given derived function. For
example, x2 has derivative 2x, so 2x has x2 as
an integral. A classic application of integral is to calculate areas. Wikipedia: Mathematics: Calculus: Integral.
Integration: The process of finding a function given its
derived function.
Intersection: The intersection of two sets is the set of
elements that are in both sets.
Intercept: A part of a line/plane cut off by another
line/plane.
Irrational
number: A real number that cannot be expressed
as the ratio of two integers, and therefore that cannot be written as a decimal
that either terminates or repeats. The square root of 2 is an example because
if it is expressed as a ratio, it never gives 2 when multiplied by itself. The
numbers p = 3.141592645...,
and e = 2.7182818... are also
irrational numbers. See also transcendental
numbers, real numbers, and Types
of Numbers.
Iteration: Repeatedly performing the same sequence of steps.
Simply, solving an algebraic equation with an arbitrary value for the unknown
and using the result to solve it again, and again.
Least
squares method: A method of fitting a straight
line or curve based one minimisation of the sum of squared differences
(residuals) between the predicted and the observed points. Given the data
points (xi, yi), it is possible to fit a straight line using a
formula, which gives the y=a+bx. The gradient of the straight line b is given
by [S(xi - mx)(yi-my)] / [(S(x-mx))2], where mx and my are the means for xi and yi. The intercept a is obtained by my - bmx. See Wikipedia: Least
Squares.
Linear: A model or function where the input and output are
proportional.
Linear
expression: A polynomial expression with the
degree of polynomial being 1, i.e., that does not include any terms as the
power of a variable. It will be something like, f(x)=2x1+3, but not
x2+2x+4 (the latter is a quadratic expression). Linear equations are closely related to a
straight line.
Literal
numbers: Letters representing numbers (as in
algebraic equations).
Logarithm: The logarithm of a number N to a given base b is
the power to which the base must be raised to produce the number N. Written as
logb N. Naturally, logb bx = x.
In any base, the following rules apply: log (ab) = log a + log b; log (a/b) = log a
- log b; log (1/a) = -log a; log ab = b log a; log 1 = 0 and log 0 is undefined.
Logistic
model (map, sequence): Wikipedia: Logistic
Map; Cut-the
Knot: JAVA
Model; Logistic
Map (interactive).